Jumps in Besov spaces and fine properties of Besov and fractional Sobolev functions

In this paper we analyse functions in Besov spaces \(B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d),q\in (1,\infty)\), and functions in fractional Sobolev spaces \(W^{r,q}(\mathbb{R}^N,\mathbb{R}^d),r\in (0,1),q\in [1,\infty)\). We prove for Besov functions \(u\in B^{1/q}_{q,\infty}(\mathbb{R}^N,\mat...

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Veröffentlicht in:arXiv.org 2024-04
Hauptverfasser: Hashash, Paz, Poliakovsky, Arkady
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Sprache:eng
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Zusammenfassung:In this paper we analyse functions in Besov spaces \(B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d),q\in (1,\infty)\), and functions in fractional Sobolev spaces \(W^{r,q}(\mathbb{R}^N,\mathbb{R}^d),r\in (0,1),q\in [1,\infty)\). We prove for Besov functions \(u\in B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d)\) the summability of the difference between one-sided approximate limits in power \(q\), \(|u^+-u^-|^q\), along the jump set \(\mathcal{J}_u\) of \(u\) with respect to Hausdorff measure \(\mathcal{H}^{N-1}\), and establish the best bound from above on the integral \(\int_{\mathcal{J}_u}|u^+-u^-|^qd\mathcal{H}^{N-1}\) in terms of Besov constants. We show for functions \(u\in B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d),q\in (1,\infty)\) that \begin{equation} \liminf\limits_{\varepsilon \to 0^+}\fint_{B_{\varepsilon}(x)} |u(z)-u_{B_{\varepsilon}(x)}|^qdz=0 \end{equation} for every \(x\) outside of a \(\mathcal{H}^{N-1}\)-sigma finite set. For fractional Sobolev functions \(u\in W^{r,q}(\mathbb{R}^N,\mathbb{R}^d)\) we prove that \begin{equation} \lim_{\rho\to 0^+}\fint_{B_{\rho}(x)}\fint_{B_{\rho}(x)} |u\big(z\big)-u(y)|^qdzdy=0 \end{equation} for \(\mathcal{H}^{N-rq}\) a.e. \(x\), where \(q\in[1,\infty)\), \(r\in(0,1)\) and \(rq\leq N\). We prove for \(u\in W^{1,q}(\mathbb{R}^N),1
ISSN:2331-8422
DOI:10.48550/arxiv.2304.09757